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is smooth formal. When it is (for example when a Q-∆ version of the ∂-∂ Lemma holds) there is a weak-Frobenius man-ifold structure on the homology of L that is important in applications and relevant to quantum correlation functions. In this paper, a necessary and sufficient condition for L to be smooth formal is presented. The condition is simply stated: it… (More)
This is the first of two papers that introduce a deformation theoretic framework to explain and broaden a link between homotopy algebra and probability theory. In this paper, cumulants are proved to coincide with morphisms of homotopy algebras. The sequel paper outlines how the framework presented here can assist in the development of homotopy probability… (More)
Let T A denote the space underlying the tensor algebra of a vector space A. In this short note, we show that if A is a differential graded algebra, then T A is a differential Batalin-Vilkovisky algebra. Moreover, if A is an A∞ algebra, then T A is a commutative BV∞ algebra.
We develop the deformation theory of A∞ algebras together with ∞-inner products and identify a differential graded Lie algebra that controls the theory. This generalizes the deformation theories of associative algebras, A∞ algebras, associative algebras with inner products, and A∞ algebras with inner products.
We describe a step toward quantizing deformation theory. The L∞ operad is encoded in a Hochschild cocyle • 1 in a simple universal algebra (P, • 0). This Hochschild cocyle can be extended naturally to a star product ⋆ = • 0+ • 1+ 2 • 2+ · · ·. The algebraic structure encoded in ⋆ is the properad Ω(coF rob) which, conjecturally, controls a quantization of… (More)
We derive an L∞ structure associated to a polarized quantum background and characterize the obstructions to finding a versal solution to the quantum master equation (QME). We illustrate how symplectic field theory (SFT) is an example of a polarized quantum background and discuss the L∞ structure in the SFT context. The discussion may be summarized as… (More)
This note defines cones in homotopy probability theory and demonstrates that a cone over a space is a reasonable replacement for the space. The homotopy Gaussian distribution in one variable is revisited as a cone on the ordinary Gaussian.